The document defines basic concepts in logic and set theory, including: - What a set is and examples like sets of numbers - Set operations like union, intersection, subsets, and complements - Properties of sets like idempotence, commutativity, associativity, and distributivity - Relations as subsets of Cartesian products, with examples of binary relations

1. Logic
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2. What exactly is logic?

3. What exactly is logic?
the study of the principles of correct reasoning

4. Wax on … wax off … these are the basics
http://www.youtube.com/watch?v=3PycZtfns_U

5. Sets
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6. computer
information information
computation

7. Set
A set is a group of objects.

8. Set
A set is a group of objects.
{10, 23, 32}

9. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }

10. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }

11. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
Ø empty set

12. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
Ø
U
empty set
universe

13. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
Ø
U
empty set
universe
Membership
a is a member of set A

14. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
10 ∈ {10, 23, 32}
Ø
U
empty set
universe
Membership
a is a member of set A

15. Set
A set is a group of objects.
{10, 23, 32}
N = {0, 1, 2, … }
Z = {… , -2, -1, 0, 1, 2, … }
10 ∈ {10, 23, 32}
-1 ∉ N
Ø
U
empty set
universe
Membership
a is a member of set A

16. Subset A⊆B
Every member of A is also an element of B.

17. Subset A⊆B
∀x:: x∈A x∈B
Every member of A is also an element of B.

18. Subset A⊆B
∀x:: x∈A x∈B
∅ ⊆ A.
A ⊆ A.
A = B A ⊆ B ∧ B ⊆ A.
Every member of A is also an element of B.

19. Subset A⊆B
∀x:: x∈A x∈B
∅ ⊆ A.
A ⊆ A.
A = B A ⊆ B ∧ B ⊆ A.
Proper subset A⊂B
A is a subset of B and not equal to B.
Every member of A is also an element of B.

20. Subset A⊆B
∀x:: x∈A x∈B
∅ ⊆ A.
A ⊆ A.
A = B A ⊆ B ∧ B ⊆ A.
Proper subset A⊂B
∀x:: A⊆B ∧ A≠B
A is a subset of B and not equal to B.
Every member of A is also an element of B.

21. Union A∪B
∀x:: x∈A ∨ x∈B
A∪B={ x | x∈A or x∈B }

22. Union A∪B
∀x:: x∈A ∨ x∈B
A∪B={ x | x∈A or x∈B }

23. Union A∪B
∀x:: x∈A ∨ x∈B
A∪B={ x | x∈A or x∈B }
A ∪ B = B ∪ A.
A ∪ (B ∪ C) = (A ∪ B) ∪ C.
A ⊆ (A ∪ B).
A ∪ A = A.
A ∪ ∅ = A.
A ⊆ B A ∪ B = B.

24. Intersection A∩B
∀x:: x∈A ∧ x∈B
A∩B={ x | x∈A and x∈B }

25. Intersection A∩B
∀x:: x∈A ∧ x∈B
A∩B={ x | x∈A and x∈B }

26. Intersection A∩B
∀x:: x∈A ∧ x∈B
A∩B={ x | x∈A and x∈B }
A ∩ B = B ∩ A.
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
A ∩ B ⊆ A.
A ∩ A = A.
A ∩ ∅ = ∅.
A ⊆ B A ∩ B = A.

27. Complements AB, A’
∀x:: x∈A ∧ x∉B
AB={ x | x∈A and x∉B }

28. Complements AB, A’
∀x:: x∈A ∧ x∉B
AB={ x | x∈A and x∉B }
A B ≠ B A.
A ∪ A′ = U.
A ∩ A′ = ∅.
(A′)′ = A.
A A = ∅.
U′ = ∅.
∅′ = U.
A B = A ∩ B′.

30. A ∩ U = A
A ∪ ∅ = A
Neutral elements

31. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements

32. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence

33. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity

34. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
Associativity

35. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
Associativity
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributivity

36. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∩ A’ = ∅
A ∪ A’ = U
Complement
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
Associativity
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributivity

37. Similar to boolean algebra
a ∧ 1 = a
a ∨ 0 = a
Neutral elements
a ∧ 0 = 0
a ∨ 1 = 1
Zero elements
a ∧ a = a
a ∨ a = a
Idempotence
a ∧ ¬ a = 0
a ∨ ¬ a = 1
Negation
a ∨ b = b ∨ a
a ∧ b = b ∧ a
Commutativity
a ∧ (b ∧ c) = (a ∧ b) ∧ c
a ∨ (b ∨ c) = (a ∨ b) ∨ c
Associativity
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
Distributivity

38. A ∩ U = A
A ∪ ∅ = A
Neutral elements
A ∩ ∅ = ∅
A ∪ U = U
Zero elements
A ∩ A = A
A ∪ A = A
Idempotence
A ∩ A’ = ∅
A ∪ A’ = U
Complement
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Commutativity
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
Associativity
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributivity

39. A ∩ U = A A ∪ B = B ∪ A
A ∪ ∅ = A
A ∩ ∅ = ∅
A ∪ U = U
A ∩ A = A
A ∪ A = A
A ∩ A’ = ∅
A ∪ A’ = U
Neutral elements
Zero elements
Idempotence
Complement
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∩ B = B ∩ A
A ∪ (B ∪ C) = (A ∪ B) ∪ C
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(A ∩ B)’ = (A’) ∪ (B’)
(A ∪ B)’ = (A’) ∩ (B’)
Commutativity
Associativity
Distributivity
DeMorgan’s

40. A ⊆ A.
A ⊆ B ∧ B ⊆ A A = B.
A ⊆ B ∧ B ⊆ C A ⊆ C
Reflexivity
Anti-symmetry
Transitivity

41. Scissors
Paper
Stone

42. Scissors
Paper
Stone
beats
beats
beats

43. Scissors
Paper
Stone
beats
beats
beats

44. Scissors
Paper
Stone
beats
beats
beats
beats Scissors Paper Stone
Scissors FALSE TRUE FALSE
Paper FALSE FALSE TRUE
Stone TRUE FALSE FALSE

45. Scissors
Paper
Stone
beats
beats
beats
beats Scissors Paper Stone
Scissors FALSE TRUE FALSE
Paper FALSE FALSE TRUE
Stone TRUE FALSE FALSE

46. Scissors
Paper
Stone
beats
beats
beats
beats Scissors Paper Stone
Scissors FALSE TRUE FALSE
Paper FALSE FALSE TRUE
Stone TRUE FALSE FALSE

47. Scissors
Paper
Stone
beats
beats
beats
beats Scissors Paper Stone
Scissors FALSE TRUE FALSE
Paper FALSE FALSE TRUE
Stone TRUE FALSE FALSE

48. beats Scissors Paper Stone
Scissors FALSE TRUE FALSE
Paper FALSE FALSE TRUE
Stone TRUE FALSE FALSE
beats = {(Scissors, Paper),
(Paper, Stone),
(Stone, Scissors)}

49. beats Scissors Paper Stone
Scissors FALSE TRUE FALSE
Paper FALSE FALSE TRUE
Stone TRUE FALSE FALSE
beats = {(Scissors, Paper),
(Paper, Stone),
(Stone, Scissors)}
beats ⊆ {Scissor, Paper, Stone} x
{Scissor, Paper, Stone}

50. Cartesian product AxB
AxB={ (a,b) | a∈A and b∈B }

51. Cartesian product AxB
AxB={ (a,b) | a∈A and b∈B }
A × ∅ = ∅.
A × (B ∪ C) = (A × B) ∪ (A × C).
(A ∪ B) × C = (A × C) ∪ (B × C).

52. N-ary Relation
A1, A2, ..., An
R ⊆ A1 x A2 x...x An

53. N-ary Relation
A1, A2, ..., An
R ⊆ A1 x A2 x...x An
Binary Relation

54. N-ary Relation
A1, A2, ..., An
R ⊆ A1 x A2 x...x An
Binary Relation
A1, A2
R ⊆ A1 x A2
(a,b) ∈ R
aRb

55. N-ary Relation
A1, A2, ..., An
R ⊆ A1 x A2 x...x An
Binary Relation
A1, A2
R ⊆ A1 x A2
(a,b) ∈ R
aRb

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